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Throughout the score there are various CX, CVII signs and the alla breve sign (which is obviously not Alla Breve). From what I could gather, these are fingering instructions for the pinky (C), but I couldn't find a full explanation.
The source can be found here:
What are these and how do I read them?
They're fret numbers. Sometimes they refer to that fret barred, sometimes (as in the sign lke cut time) they refer to a part barre. The lowest notes are usually best left played as open, although here, CV refers to fret 5, so the low A could be played on 6th string, fret 5 - BUT - since it's a half barre, it indicates play it open. The D, on the other hand (sic) might be better played fretted at 10, as indicated, making the staccato easier to perform.
The C with a slash through means a half-barre of some sort 2 - 5 strings as the harmony dictates.
The straight C probably indicates a full barre but the notation is awkward.
The position is indicated with Roman Numerals as is the tradition.
I - One
II - Two
III - Three
IV - Four
V - Five
VI - Six
VII - Seven
VIII - Eight
IV - Nine
X - Ten
XI - Eleven
XII - Twelve.
These are fret numbers, as @Tim already said, but they do not necessarily indicate full or partial barré. They indicate position. This sometimes does mean a barré, sometimes it indicates just the playing position on the fretboard.
PS: as requested in the comment: the position is indicated by a "C" followed by a roman numeral, i.e. "CIV" for fourth position or "CVII" for seventh position. It indicates the "root fret" for the hand position on the fretboard, which sometimes means to play a full or partial barré, sometimes it just indicates the finger position.
For instance: playing a am7 chord as A-x-g-c'-e'-x (5th fret, A-string and e-string muted)) would also be 5th position without any barré involved.
